# Integral Help

1. Dec 6, 2007

### silver-rose

1. The problem statement, all variables and given/known data

$$\int\frac{\arctan{x}dx}{x(x^2+1)}$$

I've been thinking over this for the past few days...I'm still stuck though
Can this integral even be expressed with elementary functions?

2. Relevant equations
N/A

3. The attempt at a solution
use the substitution $$u=tan{x}$$, and then, use integration by parts.
However I end up with $$\int\left|ln(cosx)\right|$$, as a term, which I cannot manage to integrate.

Last edited: Dec 6, 2007
2. Dec 6, 2007

### arunbg

Hint: Use u=arctan(x). What is du?

3. Dec 7, 2007

### coomast

I think this integral does not have a "classical" primitive. After the substitution and partial integration you end up with:

$$I=\frac{1}{2}\frac{arctan^2(x)}{x^2}+\frac{1}{2}\int \frac{u^2}{sin^2(u)}du$$

The remaining integral is not an elementary function, according to "the integrator" of mathematica.

@silver-rose: What is expected as a result? A classical function or an advanced one?

4. Dec 7, 2007

### Defennder

I agree. I can't find an elementary function despite a few pages of calculations and random substitutions. Maybe it's because I'm dumb or something. Anyone else had better luck here?

Tried it out at http://integrals.wolfram.com/

The answer was given in some weird notation involving something called a polylogarithm. What's that?

Last edited: Dec 7, 2007
5. Dec 7, 2007

### arunbg

Sorry my bad, I thought the OP had got it wrong in substituting u=tan(x) instead of arctan(x), and didn't check further. Since the mathematica integrator doesn't report a solution in terms of elementary functions, it is highly unlikely that there actually exists one.

6. Dec 9, 2007

### silver-rose

yea that's what i think so too..

I've spent days on this integral, basically trying tons and tons of substitutions.

ti-89 can't do it, and mathematica gives a non-elementary answer.

Thanks anyways guys.

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